HAL Id: hal-01849589
https://hal.archives-ouvertes.fr/hal-01849589
Submitted on 26 Jul 2018
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
A nonequilibrium force can stabilize 2D active nematics
Ananyo Maitra, Pragya Srivastava, M. Cristina Marchetti, Juho Lintuvuori,
Sriram Ramaswamy, Martin Lenz
To cite this version:
Ananyo Maitra, Pragya Srivastava, M. Cristina Marchetti, Juho Lintuvuori, Sriram Ramaswamy, et
al.. A nonequilibrium force can stabilize 2D active nematics. Proceedings of the National Academy of
Sciences of the United States of America , National Academy of Sciences, 2018, 115 (27), pp.6934-6939.
�hal-01849589�
A nonequilibrium force can stabilize 2D
active nematics
Ananyo Maitraa,1, Pragya Srivastavab, M. Cristina Marchettic, Juho S. Lintuvuorid, Sriram Ramaswamye,f, and Martin Lenza,g,1
aLPTMS, CNRS, Universit ´e Paris-Sud, Universit ´e Paris-Saclay, 91405 Orsay, France;bTheoretical Physics of Biology Laboratory, The Francis Crick Institute, London NW1 1AT, United Kingdom;cPhysics Department, Soft and Living Matter Program, Syracuse University, Syracuse, NY 13244;dUniversit ´e Bordeaux, CNRS, LOMA, UMR 5798, F-33405 Talence, France;eTata Institute of Fundamental Research Centre for Interdisciplinary Sciences, Hyderabad 500107, India; fCentre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, India; andgMultiScale Material Science for Energy and Environment, Unit ´e Mixte Internationale 3466, CNRS-Massachusetts Institute of Technology, Cambridge, MA 02139
Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 15, 2018 (received for review November 27, 2017) Suspensions of actively driven anisotropic objects exhibit
distinc-tively nonequilibrium behaviors, and current theories predict that they are incapable of sustaining orientational order at high activ-ity. By contrast, here we show that nematic suspensions on a substrate can display order at arbitrarily high activity due to a pre-viously unreported, potentially stabilizing active force. This force moreover emerges inevitably in theories of active orientable flu-ids under geometric confinement. The resulting nonequilibrium ordered phase displays robust giant number fluctuations that can-not be suppressed even by an incompressible solvent. Our results apply to virtually all experimental assays used to investigate the active nematic ordering of self-propelled colloids, bacterial sus-pensions, and the cytoskeleton and have testable implications in interpreting their nonequilibrium behaviors.
active matter | living liquid crystals | confined active nematics
L
iving systems convert chemical energy into motion, thus vio-lating detailed balance at the microscopic scale. Macroscop-ically, these violations result in stresses and currents responsible for intracellular flows leading to cellular motion (1), collec-tive cell migration during embryonic development (2), and the flocking of birds (3). Similar nonequilibrium currents arise in nonliving systems such as chemotactic colloids (4) and vibrated granular rods (5). These systems are often described by active hydrodynamic theories, a class of continuum descriptions derived from equilibrium theories of liquid crystals but supplemented with extra “active” forces arising from microscopic driving (6–8). These theories are tools of choice to study specifically nonequi-librium features in the collective behaviors of fluid suspensions of anisotropic active units such as cytoskeletal filaments (9, 10) or bacteria (11).A central issue in active hydrodynamics is to determine the effects of activity on the dynamic stability and the robustness against fluctuations of various types of orientational and trans-lational order. Previous studies have shown that, due to the interplay of active stress and solvent flow, nematic order in incompressible active suspensions is always unstable beyond a critical value of activity (7, 8, 12–14). This instability thresh-old vanishes in the limit of infinite system size, implying that, unlike their equilibrium counterparts, these systems are gener-ically unstable. In 2D experimental realizations this instability can, however, be suppressed by the friction of the fluid against a substrate. Nevertheless, current theories predict that even under these conditions instability always occurs at high enough activity (15), which may be experimentally realized through an increase of the density of myosin motors or bacteria or of the amount of fuel available to them.
Another distinctive feature of active systems is the statis-tics of their density fluctuations. In equilibrium systems away from critical points and with finite-range interactions, a region of space containing N particles on average will undergo fluc-tuations of this number of order √N whether or not it is
embedded in an incompressible solvent. In contrast, active hydrodynamic theories of systems without incompressibility dis-play fluctuations of order greater than√N due to active mass currents arising from orientation fluctuations (5, 7, 8, 16, 17). While these so-called giant number fluctuations have clearly been observed in solventless settings, little is known about their form in the presence of an incompressible solvent (18), and their observation in biological experiment has been difficult and controversial (19).
To help interpret the rich dynamical behavior of typically quasi-2D experiments on active systems (10, 11, 19, 20) here we study theoretically the ordering and fluctuations of an apolar active fluid in contact with a substrate. By reexamining the foun-dation of active hydrodynamic theories in two dimensions, we first find that the contact with a substrate allows an extra active force with a distinct angular symmetry. This force does not con-serve angular momentum, yet it exists even in achiral systems. Contrary to common wisdom, increased activity in the presence of this term can lead to a stabilization of nematic ordering. Here, we first qualitatively discuss the geometrical meaning of this extra force and derive its form from symmetry considerations in a purely 2D system. We next show that giant number fluctua-tions in the active nematic phase are robust to the introduction of incompressible solvent as well as this active force. Finally, we show that this potentially stabilizing active force emerges
Significance
Living systems differ from dead matter in one crucial aspect: They are driven by internal engines. In anisotropic fluids, the widely accepted framework of active matter theory predicts that this necessarily leads to an instability from quiescence to large-scale and eventually incoherent motion. Here we chal-lenge this common wisdom by reexamining the symmetries of the most ubiquitous experimental geometry for active sys-tems. We uncover an additional coupling between activity and motion that can make an active system even more stable than its passive counterpart, while preserving other hallmarks of nonequilibrium physics. Our results challenge common views on active matter and clarify the interpretation of multiple experiments.
Author contributions: A.M., M.C.M., S.R., and M.L. designed research; A.M., P.S., J.S.L., S.R., and M.L. performed research; and A.M., P.S., M.C.M., J.S.L., S.R., and M.L. wrote the paper.
The authors declare no conflict of interest. This article is a PNAS Direct Submission. Published under thePNAS license.
1To whom correspondence may be addressed. Email: ananyo.maitra@u-psud.fr or
martin.lenz@u-psud.fr.y
This article contains supporting information online atwww.pnas.org/lookup/suppl/doi:10. 1073/pnas.1720607115/-/DCSupplemental.
APPLIED
PHYSICAL
SCIENCES
naturally from an accepted model of 3D active fluid under verti-cal confinement and argue that it is the dominant active force in a renormalized theory in the presence of noise. Our results offer plausible explanations for the persistence of order at high activ-ity, as well as the systematics of instabilities, in bacterial, living liquid crystal, and cytoskeletal systems.
Qualitative Argument
The well-known destabilization of an active nematic is a result of particles exerting dipolar force distributions of either sign on the solvent surrounding them. As illustrated in Fig. 1 A and B, dipo-lar force distributions of either sign always destabilize the ordered state. The core of our argument consists of remarking that unlike these dipolar distributions, force distributions of higher angular symmetry do not generically destabilize the ordered phase. To see this, consider the square shape with quadrupolar force distribu-tion shown in Fig. 1C, which pulls fluid perpendicular to its faces and pushes it out of its corners. Perturbing a perfectly aligned row of particles exerting this more symmetric active force density leads to a bunching of their corners in some locations, which rein-forces their pushing of the fluid, and a spreading out in others. As a result, fluid tends to flow from the spread-out to the bunched-up regions, i.e., in the direction opposite to that pictured in Fig. 1A, which pushes the particles back toward the perfectly aligned state. Moreover, the fourfold symmetry of the particles prevents them from distinguishing between splay and bend perturbations. As a result, their activity—if of the right sign—always helps stabilize the aligned state.
Beyond these schematic examples, the force distribution in an active nematic generically has both dipolar and higher symme-try contributions. However, in active systems such as suspended films and 3D bulk fluids where momentum is conserved, all higher symmetry contributions are subdominant to the dipolar one at large scales. Here, we show that this is not the case in 2D systems on a substrate as well as 3D fluids under vertical con-finement. As a result, depending on its magnitude and sign the higher angular symmetry active force can lead to an overall sta-bilization of the active fluid, in contradiction with the widely held view that incompressible active nematics are always unstable at high activity.
Fig. 1. Active forces influence the stability of an orientable fluid. (A, Top) A nematic “puller” aspirates the surrounding solvent along its long direc-tion (blue arrows) to reject it along the short direcdirec-tion (red arrows). (A,
Bottom) The splayed configuration thus results in the solvent flows shown
by the thick arrows, which drag the particles along and thus accentuate the original disturbance by rotating them as indicated by the thin arrows. (B) “Pushers” with the opposite force dipole, while stable against splay, are unstable when in a bend configuration. (C) The active force intro-duced here is also present in nematic active systems, but is associated with higher multipoles of the force density distributions. In the example shown here, the resulting flow consistently stabilizes the ordered state of the par-ticles. We show that for systems on a substrate this active stabilization can overcome the well-characterized destabilization associated with the dipolar terms.
Model for Active Nematic Suspension on a Substrate
We now offer a formal description of the dynamics of the local orientation θ(r, t) and velocity v(r, t) of our nematically ordered 2D active suspension, as well as the concentration c(r, t) of its active particles.
Dynamics of the Orientation Field. Considering small deviations
from a homogeneous state aligned along ˆx, we write the linear dynamical equations compatible with the symmetries of the system in the long-wavelength (hydrodynamic) limit. The dynam-ical equation for the angle field is
˙ θ =1 − λ 2 ∂xvy− 1 + λ 2 ∂yvx− Γθ δH δθ, [1] where |λ| > 1 describes particles with a tendency to align under a shear flow (e.g., cytoskeletal filaments) while |λ| < 1 denotes flow tumbling (as in bacteria). Here we use the simplified one-Frank-constant free-energy functional
H = Z d2r K 2(∇θ) 2 + g (c) , [2]
where K > 0 characterizes the tendency of the particles to align and g(c) is an arbitrary function of the concentration.
Flow Field and Active Forces.Flow is driven by forces internal
to the fluid, and the presence of the substrate dictates Darcy dynamics,
Γv = −∇Π + fp+fa, [3]
where Γ can be viewed as the friction coefficient against the sub-strate. The pressure Π serves as a Lagrange multiplier enforcing the incompressibility condition ∇ · v = 0 for the suspension as a whole, while still permitting fluctuations in the concentration of suspended particles. We do not consider the case of a com-pressible medium, which adds no physics of interest. Onsager symmetry and Eq. 1 yield the density of passive (equilibrium) forces fp= −1 + λ 2 ∂y δH δθ ˆ x +1 − λ 2 ∂x δH δθ ˆy. [4]
Beyond these standard equilibrium terms, the active force den-sity fa depends on θ only through its gradient due to rotational
invariance. Combining this with the (x , y, θ) → (x , −y, −θ) reflection invariance of our achiral system dictates that to lowest order in gradients
fxa = −(ζ1∆µ + ζ2∆µ)∂yθ [5a]
fya = −(ζ1∆µ − ζ2∆µ)∂xθ, [5b]
where ζ1 and ζ2 are two independent, a priori unknown
phe-nomenological constants, and ∆µ denotes the strength of the overall activity in the system, e.g., the chemical potential differ-ence between the cellular fuel ATP and its hydrolysis products. We interpret the two components of fa in Fig. 2, with fxa
induc-ing a horizontal fluid flow in a splayed nematic while fa y drives a
vertical fluid flow in a bent nematic. An active force depending on gradients of c is also allowed in fa, but does not significantly
modify our discussion (SI Appendix, IA. Dynamics on a Sub-strate). While the active force proportional to ζ1 is standard in
active fluid theories, the present work introduces and explores the effect of the solenoidal ζ2force.
This force can be understood in simple terms by introducing the nematic director n = (cos θ, sin θ). In momentum-conserving systems, the active force can be only the divergence of a symmet-ric stress, namely fa= ζ
1∆µ∇ · (nn) = ζ1∆µ[n(∇ · n) + n · ∇n].
A
B
Fig. 2. The two components of the active force fadetermine the stability of
the active nematic with respect to splay and bend. (A) A splay perturbation ∂yθ 6=0 (black segments) induces an active force fxa=a∂yθwhich stabilizes
a flow-tumbling system if a < 0 (represented by the arrows). (B) A bend perturbation ∂xθ 6=0 produces an active force fya=b∂xθwhich stabilizes a
flow-tumbling system if b > 0.
The ζ2 force involves an exchange of angular momentum with
the substrate, and introducing it thus allows an active force with different prefactors for the terms n(∇ · n) and n · ∇n. The form of active force density we propose also has the same symmetry as the general flexoelectric polarization in nematic liquid crys-tals, whereas ∇ · (nn) is analogous to a more restricted case in which the magnitudes of the polarization for a bend and a splay deformation of equal strength are the same (21, 22). Since a local polarization in a system out of equilibrium may be associated with a local force density, this analogy provides a further heuristic justification for Eq. 5.
Concentration Dynamics. The evolution of the concentration c
is governed by a conservation equation ∂tc = −∇ · (Jp+Ja),
where the passive particle current reads Jp
= −Γc∇δH/δc. As
with fa, the active current Jacomprises two distinct θ-dependent
active terms a priori, but the term analogous to ζ2, being a curl,
drops out of the conservation equation, yielding ∂tc = Γc∇2
δH
δc + ζc∆µ∂x∂yθ, [6] where the ζc∆µactive term couples orientation fluctuations with
concentration fluctuations and is featured in standard theories of active nematics (7, 16).
Active Stabilization of the Ordered Phase
The active term ζ2∆µhas dramatic consequences for the
lin-ear stability of the active fluid. Consider the evolution of a small perturbation θqeiq·x with a wave vector q = q(cos φˆx +
sin φˆy). Combining Eqs. 1–6 and eliminating pressure by pro-jecting on the direction perpendicular to q, we find that the dynamics are diffusive to leading order in q: ∂tθq=
−D(φ)q2
θq, where the direction-dependent orientational
diffu-sivity is given by D (φ) = ΓθK +
∆µ
2Γ(1 − λ cos 2φ)(−ζ1cos 2φ + ζ2). [7] The system is thus linearly stable at small q if and only if D(φ) is positive for all values of φ, i.e., if the second term in the right-hand side of Eq. 7 is not so large as to overcome the stabilizing effect of director relaxation through ΓθK. We now focus on this
second term, which dominates for high activity, i.e., large ∆µ. The case of flow-tumbling (|λ| < 1) systems, where (1 − λ cos 2φ) is always positive, is easy to interpret as it is controlled by the active force fa. As shown in Fig. 2, splay can be stabilized by
a force fa
x that depends on ∂yθ only through a negative
coef-ficient, whereas bend stabilization requires fa
y to depend on
∂xθthrough a positive coefficient. As previous studies implicitly
assume ζ2= 0, Eq. 5 clearly shows that they impose the
equal-ity of these two coefficients, implying a destabilization of either bend or splay, depending on its sign. The introduction of the active force (ζ26= 0) now offers the possibility for these
coeffi-cients to have opposite signs. For ζ2> |ζ1| this implies that the
stability of the system, i.e., a positive relaxation rate, increases with increasing activity, as shown in Fig. 3A. By contrast, Fig. 3 shows that both flow-aligning (|λ| > 1) and ζ2< |ζ1| systems
remain generically unstable at high activity. Giant Number Fluctuations
Having opened up the possibility of a stable homogeneous nematic at high activity, we now examine the nature of concen-tration fluctuations arising in such a state when nonconserving and conserving noise sources are added to Eqs. 1 and 6, respec-tively. We find that the giant number fluctuations persist despite the long-range effects associated with the incompressible veloc-ity field. This result, which is in clear contrast to the case of incompressible active polar systems (23), can be seen with-out a detailed calculation by examining the structure of Eqs.
6and 7. Indeed, compared with compressible nematic systems, incompressibility introduces only a nonsingular anisotropy in the orientational relaxation rate Eq. 7, without modifying the scaling with wavenumber. Since giant number fluctuations rely solely on the wavenumber scaling of the orientational relaxation rate and not on its anisotropy (16), they should be present in our system as well. Specifically, simple power counting within a linearized treatment demonstrates that, due to the active con-centration current, concon-centration fluctuations δc = c − hci scale with q in the same way as orientation fluctuations. Therefore, the static structure factor Sq≡
R
rhδc(0)δc(r)i exp(−i q · r)/hci
of concentration fluctuations scales as 1/q2. Equivalently, in a
region containing on average N particles, the SD in the number scales linearly with N in two dimensions. Only the form of the predicted anisotropy of Sqis modified (SI Appendix, IA.
Dynam-ics on a Substrate) with respect to that in ref. 16, due to the factor D (φ)in Eq. 7. Our conclusions remain valid even upon inclusion of more general symmetry-allowed active and passive terms (SI Appendix, IA. Dynamics on a Substrate).
Emergence of the Additional Active Force in Confined Active Nematics
To further elucidate the physical origin of our active force, we consider the dynamics of a 3D active suspension of lateral
A
B
− − − − − − − −Fig. 3. Regions of stability of the ordered phase as given by the sign of
D(φ) in Eq. 7 as a function of the flow-alignment parameter λ, the ratio
ζ1/ζ2of the two active forces, and the overall magnitude of activity
rela-tive to passive friction ∆˜µ = ζ2∆µ/2KΓΓθ. (A) For ∆˜µ >0, the region of
linear stability of the ordered phase (shades of blue) shrinks with increasing activity, yet the central dark blue square is stable for arbitrary high activ-ity. (B) For ∆˜µ <0, stability is abolished for large enough activity, namely ∆ ˜µ < −1.
APPLIED
PHYSICAL
SCIENCES
dimension L confined over a length scale h L in the z direction and project it onto a 2D xy plane. To perform this opera-tion, we use a standard lubrication approximation correct in the limit of small h/L (24, 25). This approximation is based on a separation of scale between the gradients and velocities within and perpendicular to the xy plane, namely ¯∂z ( ¯∂x, ¯∂y) and
¯
vz (¯vx, ¯vy), where the bar denotes 3D variables and
opera-tors. The 2D flow equations are then obtained by averaging each term in the 3D flow equations over the thickness of the confined fluid. We illustrate this procedure by averaging the 3D viscous force density ¯∇2
¯
v found in both the standard Navier–Stokes equation and the flow equation for a full 3D active fluid. We consider a simple Poiseuille profile ¯v⊥(x , y , z ) = (4/h2)z (h −
z )¯v0(x , y ) for the vertical structure of the flow, where ¯v⊥=
(¯vx, ¯vy) is the projection of the 3D velocity vector in the xy
plane, ¯v0(x , y ) is its value in the midplane, and where a
no-slip condition is imposed on the bounding surfaces at z = 0 and z = h. Denoting by v(x , y) = (2/3)¯v0(x , y )the z -averaged
velocity we find that the thickness-averaged viscous force density reads 1 h Z h 0 ¯ η ¯∇2¯v⊥dz = − 12¯η h2 v + O " h L 2# , [8]
which gives rise to the left-hand side of Eq. 3 with Γ = 12¯η/h2.
Crucially for our subsequent discussion, we note that this aver-aged, 2D viscous force density is of zeroth order in gradient despite the fact that its 3D counterpart is proportional to ¯∇2.
Indeed, in the process of averaging each vertical gradient ¯∂z is
effectively replaced by a factor proportional to 1/h, and the hor-izontal gradients drop out because they are subdominant in the h Llimit. Although the prefactor of Eq. 8 assumes a Poiseuille profile for the flow, only the numerical coefficient in its right-hand side and thus the definition of Γ change if other kinds of flows are considered.
In 3D active fluids, this viscous force density is balanced by the divergence of active stresses. This divergence can be expressed in terms of 3D nematic order parameter ¯Q(¯r, t) as
¯ ∇ · ¯σ = ¯∇ · ¯ζ0∆µ¯I − ¯ζ1∆µ ¯Q − ¯ζ2∆µ ¯ ∇¯ Q · ¯∇ · ¯Q S . [9]
Here the superscript S denotes the symmetric part of a tensor and ¯I denotes the unit tensor. While the two first terms in the right-hand side of Eq. 9 are standard, the last term would be disregarded in a gradient-expansion treatment of an unconfined 3D fluid. Indeed, while the former are of first order in the 3D gradient ¯∇, the latter is of order three. However, similar to the averaged viscous force density of Eq. 8, its thickness-averaged counterpart gives rise to lower (first)-order terms in horizontal gradient and thus cannot be neglected. This term thus gives rise to the second activity constant ζ2discussed in previous
sections. To see this more clearly, note that one of the terms aris-ing from the force ¯∇jn ¯∇i
¯ Q · ¯∇ · ¯Q j oS is ¯∇2¯ Q · ¯∇ · ¯Q≈ ¯ ∂z2 ¯
Q · ¯∇ · ¯Q . To average this and other terms, we supple-ment the standard procedure described for the viscous force density with a mean-field treatment whereby the z average of a product of two fields is approximated by the product of their indi-vidual z averages (SI Appendix, II. Derivation of Effective Two-Dimensional Equations of Motion from Three-Two-Dimensional Equa-tions for Active Fluids Confined in One Direction). Denoting the z-averaged projection of the apolar order parameter onto the xy plane as Q(x, t) =S 2 cos 2θ sin 2θ sin 2θ − cos 2θ , [10]
the vertical average of the divergence of the third term in the right-hand-side of Eq. 9 contributes a 2D force density of the form Q · (∇ · Q) to the thickness-averaged force balance equa-tion. As this contribution is of the same order in 2D gradient as the standard active force density ∇ · (¯ζ1∆µQ), it must be
retained in a 2D description.
Performing this averaging procedure on the evolution equa-tion for the 3D nematic order parameter as well as the force balance equation (SI Appendix, II. Derivation of Effective Two-Dimensional Equations of Motion from Three-Two-Dimensional Equa-tions for Active Fluids Confined in One Direction), we obtain the full thickness-averaged 2D dynamical equations
∂tQ = −v · ∇Q + ω · Q − Q · ω − λU − ΓθH, [11a]
Γv = −∇Π − λ∇ ·H − 2∇ · (QH)A
,
−ζ1∆µ∇ ·Q − 2ζ2∆µQ · (∇ · Q), [11b]
where ζ1= ¯ζ1and ζ2= 9 ¯ζ2/h2. The thickness-averaged
dynam-ics of the concentration field are described by Eq. 6. These equations reduce to Eqs. 1–5 deep in the ordered phase where S relaxes within a microscopic time to its steady-state value, which we set to 1 without loss of generality. In Eq. 11 H is the molecular field conjugate to Q, the superscript A denotes the antisymmetric part of a tensor, and ω and U are, respectively, the antisymmetric and symmetric parts of the tensor ∇v. The pressure Π imposes incompressibility as in Eq. 3. Eq. 11 demonstrates that the force ∝ ζ2previously introduced through general symmetry arguments
is a natural emergent feature of confined 3D active dynamics. Note that while we have included only one term of order ¯∇2
in Eq. 9 for clarity, other O( ¯∇n≥2)terms also contribute both
to this extra ζ2 active force and to the usual ζ1one upon
thick-ness averaging and do not introduce any qualitatively new term. Dimensionally, each additional factor of ∇ in the 3D theory must be accompanied by a prefactor of order `, a length scale given by the size of the suspended particles. Meanwhile, similar to Eq. 8 each vertically averaged gradient yields a factor 1/h. We thus expect ζ2/ζ1∼ (`/h)2. As a result, in closely confined
suspen-sions with h ∼ ` the coefficients ζ2and ζ1should be comparable
in magnitude.
Dominance and Generality of the Extra Active Force
Beyond these microscopic considerations, we predict that the ζ2 force will dominate over the old ζ1 force in a
renormal-ized theory in the presence of noise. Indeed, according to Eqs.
10and 11b the latter reads ∇ · Q = cos 2θ(∂yθˆx + ∂xθˆy )in the
ordered phase, while the former takes the form 2Q · (∇ · Q) = ∂yθˆx − ∂xθˆythat does not involve the anisotropic factor cos 2θ.
Since active nematics have only quasi–long-range order, all anisotropic terms average to zero at large scales due to rota-tion invariance (17), implying that hcos 2θi decays as a power of system size with a typically small exponent. Therefore, for large systems, the isotropic ζ2active force does not vanish with
diverging system size (SI Appendix, III. Beyond Linear Theory: Scaling of Active Force with Distance) and thus dominates over the ζ1force.
In addition to its role in nematic and polar systems, the active force introduced here is the key to characterizing activity in higher-symmetry active systems, including tetractic (5) and hex-atic (26, 27) phases, as expected from the schemhex-atic of Fig. 1C. In these systems symmetry imposes that ζ1, ζc, and λ in Eqs.
1–6 all vanish, implying that our ζ2 term is the only possible
source of active instabilities. It arises through an antisymmet-ric piece of the active stress, proportional in two dimensions to θ in the θ 1 limit, where the pseudoscalar angle field θ is the broken-symmetry mode and is the 2D Levi–Civita
tensor. While this does not give rise to a generic instability, these active p-atics on substrates are nevertheless unstable for ζ2∆µ < −2ΓΓθKirrespective of the presence of an
incompress-ible solvent. In addition, the aforementioned pure-curl character of the ζ2 term means that they do not contribute to mass
cur-rents. Active tetratics and hexatics thus have normal (nongiant) number fluctuations.
Dynamics of Defects
Beyond linear stability analysis, activity in orientable fluids also modifies the dynamics of defects. In nematics, the most abundant defects have charges +1/2 and −1/2. It is well known that +1/2 defects are motile in active nematics (5, 28–30). The ζ2 active
force does not lead to any additional propulsion of these defects. Indeed, the force density generated by this term is isotropic, (ζ2∆µ/2r )ˆr, and thus cannot lead to defect propulsion.
Simi-larly, in systems with higher symmetry where only the ζ2 term
is allowed, defects parameterized as θ = ±nψ in polar coordi-nates (r , ψ) (with n = 1/4 for tetratics and n = 1/6 for hexatics) lead only to an isotropic active force density. Therefore, bal-listic motion of defects is impossible in such higher symmetry systems.
Experimental Consequences and Discussion
The presence of the ζ2active term has measurable implications
for current experiments on biological active matter. First and foremost, we predict the existence of stable states for arbitrarily high values of the activity parameter ∆µ. This helps rational-ize the recent observation of highly ordered apolar nematic phases in confined suspensions of Escherichia coli bacteria (19). The absence of bacterial turbulence in these systems would have been a puzzle in a treatment with a single activity param-eter ζ1. Indeed, the experiments use a nontumbling mutant,
i.e., a system with a highly reduced Γθ in Eq. 7, which should
favor instability. The resolution could well lie in our mecha-nism involving a second activity parameter with a possibly stabi-lizing effect.
While this qualitative prediction helps explain the persistence of stable states in situations with high activity or low dissipa-tion, it is not sufficient to determine whether the aligned state in a given experiment is stable due to our mechanism or because the largest experimentally accessible ∆µ is simply too small. To discriminate between these two possibilities, we note that in a noisy environment the stabilizing influence of the ζ2active term
on the diffusivity D(φ) given in Eq. 7 is directly reflected in the magnitude of the angular fluctuations even within a stable aligned state. Considering an angular noise strength 2T Γθ, our
theory thus predicts that the equal-time correlator for angular fluctuations reads
hθq(t )θ−q(t )i =
T Γθ
D (φ)q2. [12]
In a theory including only the old ζ1 active term, an increase
of the ratio ∆µ/Γ (e.g., through an increase of the amount of ATP available in a cytoskeletal assay) should lead to a decrease of this correlator along certain directions φ and an increase along others. By contrast, in the presence of the stabilization mechanism discussed here (i.e., for ζ2> |ζ1|), this correlator will
decrease along all directions φ, providing a quantitative test for the existence of the ζ2stabilization.
Our linear stability analysis also helps to heuristically under-stand pattern formation in bacterial and cytoskeletal active flu-ids (10, 11, 31, 32). The anisotropy of D(φ) in Eq. 7 implies that the ordered phase is destabilized when it first becomes negative for any angle φu. In this case, bands with the
nor-mal vector oriented along φu are likely to form whose length
scale can be obtained simply by extending, to fourth order in
the wavenumber q, the 2D mode analysis that led to Eq. 7. The resulting dynamics read ∂tδθq= −[D (φ)q2+ Kr(φ)q4]δθq,
where the stabilizing coefficient Kr(φ) = (K /4Γ)(1 − λ cos 2φ)2
arises simply from Frank elasticity and thus accounts for patterns with size ≈pKr(φu)/|D (φu)| without resorting to
previously invoked ad hoc high-order gradient expansions (11, 32, 33). Moreover, when the effective 2D dynamics are those of a confined 3D fluid as discussed above, Γ ∝ 1/h2
so that Kr dominates over such ad hoc terms by a factor
O(h/`)2, which can be large depending on the scale of the confinement.
Beyond these applications, this order-q4 theory can be
extended to explain ordering and pattern formation in novel “liv-ing liquid crystals,” namely passive nematic liquid crystals with well-characterized physical properties perfused with small quan-tities of bacteria. Because of the small amount of active bacteria in them, these systems share most of their passive properties with the embedding, often well-characterized liquid crystal. This offers unprecedented opportunities to quantitatively test our and other theoretical predictions. The detailed connection between these systems and the current study is discussed inSI Appendix, IV. Living Liquid Crystals, where we introduce two coupled angle fields for the local alignment of the passive liquid crystalline par-ticles and active bacterial ones. Our approach accounts for activ-ity without the need for previously introduced ad hoc Onsager symmetry-breaking orientational couplings between these two angular fields (20) and leads to a stability criterion identical to that of Eq. 7. As a quantity analogous to λ can be quantita-tively tuned by modifying the passive properties of the liquid crystal, our stability prediction is directly testable, and we pre-dict the appearance of patterns with a similarly tunable length scale. Changing the embedding liquid crystal, the viscosity of the solvent, or the scale of the confinement should also enable mod-ifications of the parameters analogous to K , Γ, and Γθin Eq. 7
and allow direct tests of whether a flow-tumbling system with a large and positive ζ2remains stable when ∆µ/(K ΓΓθ) > 2, as we
predict.
Finally, our approach suggests an explanation for the recent observation (34) that the transition to activity-driven turbulence in extensile microtubule–kinesin systems confined in a channel is controlled by the aspect ratio of the transverse cross-section of this channel. Averaging the dynamics over the z direction in a channel with a rectangular Ly× h cross-section, we look at the
longest wavelength splay fluctuations in the y direction, while noting that extensile particles imply ¯ζ1> 0, which stabilizes splay
fluctuations. Assuming a destabilizing ¯ζ2< 0, we predict that the
system should become unstable as h is decreased and the ratio ζ2/ζ1∝ 1/h2 subsequently increases. We therefore predict that
ζ2active force plays a destabilizing role in microtubule–kinesin
systems.
ACKNOWLEDGMENTS. We thank Oleg Lavrentovich for pointing out the similarity between the active force and flexoelectric polarization. S.R. and M.C.M. thank the Kavli Institute for Theoretical Physics (KITP) for hospi-tality during completion of some of this work. This work was supported by Marie Curie Integration Grant PCIG12-GA-2012-334053, “Investisse-ments d’Avenir” Laboratoire d’excellence Physique:Atomes Lumi `ere Mati `ere (LabEx PALM) (ANR-10-LABX-0039-PALM), Agence Nationale de la Recherche (ANR) Grant ANR-15-CE13-0004-03, and European Research Council Groupement de Recherche Starting Grant 677532 (to M.L.). M.L.’s group belongs to the CNRS consortium Physique de la cellule au tissu Ini-tiative d’excellence. S.R. acknowledges support from a J. C. Bose National Fellowship of the Science and Engineering Research Board, India and from the Tata Education and Development Trust. M.C.M. was supported by the US National Science Foundation Awards NSF-DMR-1609208 and NSF-DGE-1068780, by the Simons Foundation Targeted Grant 342354, and by the Syracuse Soft Matter and Living Matter Program. A.M., P.S., S.R., and M.C.M. also acknowledge the support of the Kavli Institute for The-oretical Physics under Grant NSF PHY11-25915. S.R. and M.C.M. also thank the KITP for support under Grant PHY-1748958. J.L was supported by the Initiative d’Excellence Bordeaux Junior Chair.
APPLIED
PHYSICAL
SCIENCES
1. Kruse K, Joanny JF, J ¨ulicher F, Prost J (2006) Contractility and retrograde flow in lamellipodium motion. Phys Biol 3:130–137.
2. Hakim V, Silberzan P (2017) Collective cell migration: A physical perspective. Rep Prog
Phys 80:076601.
3. Ballerini M, et al. (2008) Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proc Natl Acad
Sci USA 105:1232–1237.
4. Saha S, Golestanian R, Ramaswamy S (2014) Clusters, asters, and collective oscillations in chemotactic colloids. Phys Rev E 89:062316.
5. Narayan V, Menon N, Ramamswamy S (2007) Long-lived giant number fluctuations in a swarming granular nematic. Science 317:105–108.
6. Ramaswamy S (2017) Active matter. J Stat Mech 2017:054002.
7. Marchetti MC, et al. (2013) Hydrodynamics of soft and active matter. Rev Mod Phys 85:1143–1189.
8. Toner J, Tu Y, Ramaswamy S (2005) Hydrodynamics and phases of flocks. Ann Phys 318:170–244.
9. Schaller V, Weber C, Semmrich C, Frey E, Bausch AR (2010) Polar patterns of driven filaments. Nature 467:73–77.
10. Sanchez T, Chen D, DeCamp SJ, Heymann M, Dogic Z (2012) Spontaneous motion in hierarchically assembled active matter. Nature 491:431–434.
11. Dunkel J, Heidenreich S, B ¨ar M, Goldstein RE (2013) Minimal continuum theories of structure formation in dense active fluids. New J Phys 15: 045016.
12. Simha RA, Ramaswamy S (2002) Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles. Phys Rev Lett 89:058101.
13. Voituriez R, Joanny JF, Prost J (2005) Spontaneous flow transition in active polar gels.
Europhys Lett 70:404–410.
14. Marenduzzo D, Orlandini E, Cates ME, Yeomans JM (2007) Steady-state hydrodynamic instabilities of active liquid crystals: Hybrid lattice Boltzmann simulations. Phys Rev E 76:031921.
15. Kumar A, Maitra A, Sumit M, Ramaswamy S, Shivashankar GV (2014) Actomyosin contractility rotates the cell nucleus. Sci Rep 4:3781.
16. Ramaswamy S, Simha RA, Toner J (2003) Active nematics on a substrate: Giant number fluctuations and long-time tails. Europhys Lett 62:196–202.
17. Shankar S, Ramaswamy S, Marchetti MC (2018) Low-noise phase of a two-dimensional active nematic system. Phys Rev E 97:012707.
18. Mishra, S (2008) Dynamics, order and fluctuations in active nematics:
Numeri-cal and theoretiNumeri-cal studies. Available at
www.openthesis.org/documents/Dynamics-order-fluctuations-in-active-601122.html. Accessed August 20, 2017.
19. Nishiguchi D, Nagai KH, Chat ´e H, Sano M (2017) Long-range nematic order and anomalous fluctuations in suspensions of swimming filamentous bacteria. Phys Rev
E 95:020601.
20. Zhou S, Sokolov A, Lavrentovich OD, Aranson IS (2014) Living liquid crystals. Proc Natl
Acad Sci USA 111:1265–1270.
21. Prost J, Marcerou (1977) On the microscopic interpretation of flexoelectricity. J Phys
France 38:315–324.
22. Meyer RB (1969) Piezoelectric effects in liquid crystals. Phys Rev Lett 22:918–921. 23. Bricard A, Caussin JB, Desreumaux N, Dauchot O, Bartolo D (2013) Emergence of
macroscopic directed motion in populations of motile colloids. Nature 503:95–98. 24. Stone HA (2002) Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO
Science Series C: Mathematical and Physical Sciences, eds Berestycki H, Pomeau Y (Kluwer Academic, Dordrecht, The Netherlands), Vol 569.
25. Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev
Mod Phys 69:931–980.
26. Redner GS, Hagan MF, Baskaran A (2013) Structure and dynamics of a phase-separating active colloidal fluid. Phys Rev Lett 110:055701.
27. Palacci J, Sacanna S, Steinberg AP, Pine DJ, Chaikin PM (2013) Living crystals of light-activated colloidal surfers. Science 339:936–946.
28. Giomi L, Bowick MJ, Ma X, Marchetti MC (2013), Defect annihilation and proliferation in active nematics. Phys Rev Lett 110:228101.
29. Pismen L (2013) Dynamics of defects in an active nematic layer. Phys Rev E 88: 050502.
30. Giomi L, Bowick MJ, Mishra P, Sknepnek R, Marchetti MC (2014) Defect dynamics in active nematics. Philos Trans R Soc A 372:20130365.
31. Giomi L (2015) Geometry and topology of turbulence in active nematics. Phys Rev X 5:031003.
32. Oza AU, Dunkel J (2016) Antipolar ordering of topological defects in active liquid crystals. New J Phys 18:093006.
33. Srivastava P, Mishra P, Marchetti MC (2016) Negative stiffness and modulated states in active nematics. Soft Matter 12:8214–8225.
34. Wu K-T, et al. (2017) Transition from turbulent to coherent flows in confined three-dimensional active fluids. Science 355:eaal1979.
